Animating a Looping Soft Body Character
Physics have a start and end cycle, there is no way to "reset" the cycle once it starts calculating. My goal was set based on a limited knowledge of how the physics work. The best way to go about my goal of making a looping gif, with the wool moving, is to add extra bones and weight the wool to the bones, and animate it manually.

Alternatively, if the pose/animation allows for it, when creating the gif, reuse the images in reverse for the second half, to make the illusion that the wool is going back into place. This doesn't work for some animations, but it'll work for a basic standing position at least.